Theorem fvopab4ndm 7030

Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Hypothesis

Ref	Expression
fvopab4ndm.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Assertion

Ref	Expression
fvopab4ndm	⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm

Step	Hyp	Ref	Expression
1		fvopab4ndm.1	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2	1	dmeqi 5901	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3		dmopabss 5915	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
4	2, 3	eqsstri 4007	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
5	4	sseli 3968	. 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴)
6		ndmfv 6927	. 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
7	5, 6	nsyl5 159	1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∅c0 4318  {copab 5205  dom cdm 5672  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-dm 5682  df-iota 6495  df-fv 6551

This theorem is referenced by:  fvmptndm  7031